Is there a googol of anything?

21 October 2012

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Question

Hi Chris, I hear there is a maths radio show this weekend so the Spriggs family has a maths question for you (it may also have some physics in it...!): A concept that we enjoy in the Spriggs household is that of the googol, i.e. 10 to the power 100 or a 1 followed by a hundred 0s. I believe that the googol may be a pretty useless concept, if there isn't a googol of anything in the universe... I said this to my boys and added that you don't need a number to describe something that doesn't exist... Since that day, when our older son was about 7,our two boys have been trying to come up with a googol of something. They started with the grains of sand in the Sahara Desert. That is a very very small number. We eventually reached the number of photons in the universe. I don't know the answer to this, but I got to 10 to the power 87. Which leaves it 10 trillion times smaller than a googol. But I have never been able to prove this. It would be good if the Naked Scientists could oblige?! And could they get us past a googol of something, anything? If nothing else, it has taught our boys the laws of indices! Best regards, Andy

Answer

Matt - A googol is a great number in that, it is a number that exists just because it has a nice name. There's nothing that we really have a googol of, but someone thought, "wouldn't a 1 with 100 zeros be a great number?" so they called it a googol. You could hear him emphasising gooGOL because it's spelled G O O G O L. The Google search engine is named after this but they spelled it differently. He's absolutely correct. If you're going to try to count objects, you won't find a googol of anything. So, even going right up to number of protons in the universe, which I think is - physicists argue between 1079 to 1081, so it's a 1 with somewhere between 79 and 81 zeros. But I have found one thing which we have more than a googol of in the Universe.

Chris - Well, should we ask Ginny if she can guess what it is?

Ginny - I don't know. I mean, I was thinking about neurons because they make an awful lot of connections, but I think when we are thinking about this, it probably wouldn't be quite that many, even if you include all the animals on the Earth.

Chris - 1011 nerve cells in the brain, 1,000 connections per nerve cells per brain, so that would probably be about 1014; so we're still a long way short of Matt's googol...

Matt - Actually, you're not far off from what I decided on, because when you're talking about connections, well, suddenly, you're looking at different ways of combining things. And so I looked at how many ways you can shuffle a pack of cards. So if you got a pack of 52 cards and you shuffle it, you can work out there's about 1067, It's 8 times 1067 ways to shuffle a pack of cards, which is a lot more than the number of grains of sand in the Sahara. In fact, if you've got a pack of cards with just 27 cards, you would have about 1028 different ways to shuffle them. If you want a googol, if you get a pack of cards with 70 cards - so you're going to need your normal pack and then get 18 from a pack maybe with a different colour on the front or the back so you can tell them apart - if you shuffle 70 cards, there's a googol of possible ways to arrange them!

Chris - Beautiful. I was thinking of a similar example, Matt, because I was looking at the power of passwords on the internet. Someone was asking a question on our forum and they were discussing the power of encryption and saying, "if I took a brute-force attempt to crack a password just by taking random choices, how many possible combinations of a password encrypted with a certain number of bits of encryption?" And you end up with something which will take trillions of centuries to solve and the number of solutions I think is way more than a googol.

Matt - Yes, and again, you're looking at number of possible combinations. So in fact, a password is pretty close to a pack of cards because you've got the alphabet, upper and lower case and then you do actually get extra symbols, and bits and pieces. So, if you've got a password of even just 8 characters, you're already up in the very high tens of thousands and much bigger than that, you're into millions, billions and well off very quickly!

Comments

It is theorized that there are a maximum 10 billion googols of just atoms in the universe. Protons, quarks, subatomic particles, etc. doesn't change much. Even if there were 100 subatomic particles per atom, the number would only go from 10^110 to 10^112.

The reason for this estimate is that the observable universe has been estimated to contain 10^86 atoms, maximum. Meanwhile, a calculation of the expansion of the universe estimated that the whole universe could be 150 sextillion (1.5*10^23) times the size of the observable universe, assuming that at the near instant of the 'big bang', expansion occurred at the exact rate that is predicted by mathematical models.

This is definitely an upper bound of the amount of atoms in the universe, but it is not unreasonable to hypothesize that there are a googol atoms in the universe. However, t is not reasonable to hypothesize that there are a googolplex atoms in the universe. This would imply that if one travelled at one decillion times the speed of light, they would continue to pass through universe after universe after universe, all with the exact same physical properties, until near the end of time. There is no evidence that there are multiple universes that contain atoms, and it is more reasonable to assume that another universe would have completely different everything. Hypothesizing otherwise is more sketchy than all the math I am using...

There is hope though, to find something that is actually a googolplex in number. While Vsauce estimated that (with my modifications) 3.5*10^250 thoughts could be thought from the beginning to the end of the universe, and that could be off by hundreds of exponents, and hundreds more, it still doesn't come anywhere close. Our most promising possibility is the total number of combinations of all subatomic particles in the space of the estimated universe. This more than covers all of the possible thoughts that could be thought in our style of thinking - as each thought is a combination of set structures of particles that form neurons. This number can never be estimated, but once again is our best bet at a googolplex number of things (things that we know to exist in a space that we also know to exist).

Correction - Permutations, not combinations. Order matters

The universe is vast. It has no boundaries. It is endless. It is infinite. Therefore, there is no limit of space. There is room for googolpex galaxies and solar systems with room to spare. Infinite space. Endless space.

Some bold statements you've made there - do you have some evidence to support those claims?

Our planet is very much tiny no living thing just like bacterias, viruses, insects, animals, plants, trees, leafs, fruits, vegetables, seeds, etc. and non living things just like grains of sand, atoms, neutron, proton, nucleas etc cannot reach the figure of the one googolplex.

With respect, did you actually read or listen to the answer above?

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